Subsurf on a crown. How can I smooth some edges and keep others sharp?
Another approach:




Create a Z-scaled, subduvided UV sphere, and a capless cylinder, at the origin. 
Give the cylinder a few horizontal loops,  and assign it a Srinkwrap modifier to the sphere; drag it down to a good-looking position in Z
Select the N,S,E,W vertices on the top loop of the Cylinder, and with Proportional Edit / Linear falloff, drag the points up in Z
Proportional off, Median Pivot Point, scale the bottom loop to 0 in Z
Assign a Solidify modifier, with 'Crease' set to 1 for Inner, Outer, and Rim.


The crease will hold against any subdivision.. after applying the modifiers you can extrude down the cylindrical headband, etc. (You might not even need subdivision, maybe AutoSmooth will do..)

EDIT: .... But as moonboots points out, it's worth taking a little time to put in a bevel after applying the Solidify and a level of subdivision around the rims, and then across the peaks and troughs, as shown: